calculus-2-assignment-4.pdf - Jacobs University Bremen Prof...

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Jacobs University Bremen Fall Semester 2017 Prof. Dr. T. Preusser Calculus II Assignment Sheet 4. Due: Nov 22, 2017 Problem 10 [10 Points]: In class we have discussed substitution and integration by parts as techni- ques that can be used for computing integrals. Another useful technique that helps in particular for integrating rational functions p ( x ) /q ( x ) is the method of partial fractions : 1. If the degree of p ( x ) degree of q ( x ) divide the polynomials to obtain p ( x ) q ( x ) = s ( x ) + t ( x ) q ( x ) , where s ( x ) , t ( x ) and q ( x ) are polynomials and degree of t ( x ) < degree of q ( x ). 2. Factor the denominator q ( x ) into linear (or irreducible quadratic) factors, i.e. write q ( x ) as the product of x - a i and x 2 + b i x + c i terms where a i is a root of q ( x ). Thus we arrive at a representation q ( x ) = ( x - a 1 ) k 1 · · · ( x - a l ) k l ( x 2 + b 1 x + c 1 ) m 1 · · · ( x 2 + b n x + c n ) m n . (2) ( This is a hard part for which there is no general recipe. ) 3. For the case k i = m i = 1 every linear factor ( x - a i ) is producing a term A i x - a i , and every quadratic factor ( x 2 + b i x + c i ) produces a term B i x + C i x 2 + b i x + c i . Terms in (2) that have powers k i , m i > 1 are more di ffi cult to treat and not covered here. 4. We have arrived at a decomposition p ( x ) q ( x ) = s ( x ) +
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  • Fall '17
  • Tobias Pressuer
  • Fraction, Rational function

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